Question

1.Historical records show that the average per capita consumption of eggs in the US is 278 (this is per year). Lately, consumers have been getting conflicting advice about the benefits and dangers of eggs, and a rural agricultural association wants to know if the average egg consumption has changed. A sample of 50 adults yielded a sample mean egg consumption of 287 eggs and a sample standard deviation of 28 eggs. Construct a null and alternate hypothesis that the association can use to answer their question. Based on the sample data and a 5% level of significance, what can they conclude?

2. In 2010, the average weight of cell phone batteries was 110 grams. Battery technology has continued to improve and a consumer group wants to know if the average weight of cell phone batteries has reduced. A sample of 49 cell phone batteries yielded a sample mean of 108 grams. Historical data shows that the population standard deviation is 10 grams. At the 5% level of significance, can the consumer group conclude that batteries have become lighter? Would your answer change at a 1% level of significance? Justify.

Answer 1

1.

Let, X:  average per capita consumption of eggs in the US

Then, where,

The rural agricultural association wants to know if the average egg consumption has changed.

Here, we consider the test:

Now, sample mean= sample size = sample standard deviation= ;

level of significance= 5%

Here, we apply a Student's t-test and use the test statistic

In order to set confidence limits to , we see that

Clearly, population mean of average per capita consumption in US is not included in the above confidence limit, i.e. it is safe to conclude that the average egg consumption in US has changed significantly.

2.

Let, X:   the average weight of cell phone batteries

Then, where,

Battery technology has continued to improve and a consumer group wants to know if the average weight of cell phone batteries has reduced.

Here, we consider the test:

Here H₀ is rejected if for the given sample the value of the statistic is smaller than

Now, sample mean= sample size = level of significance= 5%

Here, we use the test statistic

[Under null hypothesis H₀ ]

For level of significance 5% i.e.   ; , which is smaller than observed value of s. Hence, we fail to reject the null hypothesis and conclude that the batteries has not become significantly lighter based on the sample.

For level of significance 1% i.e.   ; , which is smaller than observed value of s. Hence, we fail to reject the null hypothesis and conclude the same as before.